Dispensers for bath tissue used in institutional and public restrooms are often configured to provide a “jumbo” roll of sheet material or bath tissue therein. For increased capacity and to extend the time between restocking, some of these dispensers are configured to accommodate a second, or backup, jumbo roll, which is disposed alongside the primary jumbo roll. It would be advantageous to have a dispenser that would accommodate a backup jumbo roll, yet restrict a user from dispensing sheet material from the backup roll prior to substantial depletion of the primary roll.